notation of directions in trigonal symmetry

[Robert Krakow]

Hey everyone,

I was trying to build a few pole figures for corundum for comparison. I used a three digit notation for the definition of direction in the trigonal system.
I am puzzled by the (4 digit) notation of directions for pole figures that Mtex is returning.. Usually they should follow:
U = 2u -v
V = 2v - u
T = - (u+v)
W = 3w
for a given 3 digit notation <uvw>. The notation for images 2-4 and 6 do not make sense to me.
uvw     Mtex output  expected output
001     0001              0001
100     1000              2-1-10
010     0100              12-10
110     00-10             11-20
210    10-10              10-10
551    22-31              55 -10 3

Interestingly, when comparing to pole figures with planes as poles, they match the expected result (when compared with crystal maker) and the notation is also fine.
Of course the hkl --> hkil conversion is different from the one above.

I used the symmetry 'spacegroup' 167 but the same result was obtained using '-3m'.
The code is available below. Used Mtex 3.5 btw.

Would be grateful for any help to elucidate where I asked for the wrong output or if it is a different notation or else.

 Cheers,
 Robert

-------------------------------------------------------------------------------------------------------
{%% plot specific Euler triplets in pole figure

cs = symmetry('spacegroup',167,[4.7617 4.7617 12.9947], 'Y||a*', 'X||b', 'Z||c');
orientation1 = orientation('Euler',0*degree,0*degree,0*degree,cs,'Bunge'); % standard orientation
orientation2 = orientation('Euler',264*degree,0*degree,0*degree,cs,'Bunge'); % dummy
orientation3 = orientation('Euler',264*degree,62*degree,0*degree,cs,'Bunge'); % lower grain
orientation4 = orientation('Euler',264*degree,62*degree,31*degree,cs,'Bunge'); % lower grain
m1 = Miller(0,0,1,'direction');
m2 = Miller(1,0,0,'direction');
m3 = Miller(0,1,0,'direction');
m4 = Miller(1,1,0,'direction');
m5 = Miller(2,1,0,'direction');
m6 = Miller(5,5,1,'direction');
figure
plotx2east
plotpdf(orientation1,[m1,m2,m3,m4,m5,m6],cs,'antipodal','grid')
hold on
plotpdf(orientation2,[m1,m2,m3,m4,m5,m6],cs,'antipodal','grid')
plotpdf(orientation3,[m1,m2,m3,m4,m5,m6],cs,'antipodal','grid')
plotpdf(orientation4,[m1,m2,m3,m4,m5,m6],cs,'antipodal','grid')
hold off
annotate([xvector,yvector],'label',{'x','y'},'BackgroundColor','w');
title('pf -3m symmetry and orientations 1-4, Y||a*')}

[Ralf Hielscher]

Dear Robert Krakow,

there are two things to mention

1) When defining crystal directions by Miller indices always pass the crystal symmetry as an argument, i.e.,

cs = symmetry('spacegroup',167,[4.7617 4.7617 12.9947], 'Y||a*', 'X||b', 'Z||c');

m1 = Miller(5,5,1,'direction',cs);

2) The above command still does not give the right result, as MTEX up to version 3.5 always assumes UVTW notation for trigonal and hexagonal symmetries. If "T" is not present MTEX sets it to T = -(U+V)  but does not apply the whole convention from uvw to UVWT. This means both next input give you the desired result

m2 = Miller(5,5,3,'direction',cs)
m3 = Miller(5,5,-10,3,'direction',cs)

I agree that this is confusing and, hence, it will change starting with MTEX 4.0.

I hope this helps,

Ralf.

[Robert Krakow]

Dear Ralf,

thank you for your feedback. I made alterations as suggested by you, by adding 'cs' to the Miller definition (for either case: 'direction' and 'pole'). In fact for plane poles it did not make a difference.

I also changed the definition of axes in order for them to agree with my expected result.

The only pole ('direction') that is still off is the [551] direction (last sub image).

I know from crystal maker, that [551] must be in the centre, as shown in second image.

The another observation that bothers me is that plane poles and directions for a given Miller definition ({100} vs [100]) now point in the same direction.

That is unexpected. For comparison I uploaded two images for the same orientation as output by crystal maker.

It is easy to find that the orange (100) plane pole is parallel to the green [210] direction.

I assume there are different definitions being used in Mtex as compared to Crystal maker...?

Any remarks are welcome.

 Thanks,
 Robert

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​Again, I used Mtex 3.5.

----------------------------------------------------

cs = symmetry('spacegroup',167,[4.7617 4.7617 12.9947], 'X||a*', Y||b', 'Z||c');

orientation1 = orientation('Euler',0*degree,0*degree,0*degree,cs,'Bunge'); % standard orientation
orientation2 = orientation('Euler',264*degree,0*degree,0*degree,cs,'Bunge'); % dummy
orientation3 = orientation('Euler',264*degree,62*degree,0*degree,cs,'Bunge'); % lower grain
orientation4 = orientation('Euler',264*degree,62*degree,31*degree,cs,'Bunge'); % lower grain

%dp = 'pole';
dp = 'direction';
m1 = Miller(0,0,1,dp,cs);   %001
m2 = Miller(1,0,0,dp,cs);   %100
m3 = Miller(0,1,0,dp,cs);   %010
m4 = Miller(1,1,0,dp,cs);   %110
m5 = Miller(2,1,0,dp,cs);   %210
m6 = Miller(5,5,1,dp,cs);   %actual pole in centre

figure
plotx2east
plotpdf(orientation1,[m1,m2,m3,m4,m5,m6],cs,'antipodal','grid')
hold on​
plotpdf(orientation2,[m1,m2,m3,m4,m5,m6],cs,'antipodal','grid')
plotpdf(orientation3,[m1,m2,m3,m4,m5,m6],cs,'antipodal','grid')
plotpdf(orientation4,[m1,m2,m3,m4,m5,m6],cs,'antipodal','grid')
hold off
annotate([xvector,yvector],'label',{'x','y'},'BackgroundColor','w');

title('pf -3m symmetry and orientations 1-4, X||a*')

​

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